Function estimation is a branch of mathematics having many real world applications. For example, function estimation may be applied to a finite number of noisy measurements (or "observations") for the purpose of approximating a function that describes the phenomenon being measured. Existing techniques of function estimation generally use the expansion: ##EQU1##
where f(x) is the approximating function, .phi..sub.i (s) are a plurality of a priori chosen functions, .alpha..sub.i are scalars, and "N" is the number of terms in the expansion. Accordingly, to estimate a function using equation (1), one must estimate the scalars .alpha..sub.i. The equation may be readily applied to one or two dimensional data, however its application to data of three or more dimensions is limited by the "curse of dimensionality".
The curse of dimensionality is well known among mathematicians. It refers to the exponential increase in complexity of the estimating function that occurs as the dimensionality of the data being estimated increases. For example, if a one dimensional curve may be estimated with reasonable accuracy by using 20 terms (N=20), then the estimation of a two dimensional surface with reasonable accuracy requires 20.sup.2 terms (N=20.sup.2). A three dimensional surface requires 20.sup.3 terms, and so on. Thus, as the dimension of the function to be estimated increases, the number of terms required rapidly reaches the point where even the most powerful computers can not handle the estimation in a timely manner.
Positron Emission Tomography (PET) is a graphical method of recording the physiological processes of the body. A tracer labelled with a positron emitting nuclide is injected into the body, and an image of the distribution of radioactivity which follows is reconstructed from the measured positron emissions.
There are two classes of image reconstruction algorithm used in PET. Direct analytical methods because they require relatively short computation times. Images reconstructed by analytical methods tend to have low visual quality and display interference between regions of low and high tracer concentration. Iterative methods generally produce higher visual quality, but are computationally much more intensive. At present, the computation time of reconstruction using iterative methods is unacceptably long.
At present, analytical and iterative methods require the problem to be discretized at some stage. This means that the activity concentration is calculated at a finite number of points from a finite number of projections. The area to be reconstructed is subdivided into a mesh of pixels, and the activity concentration within each pixel is assumed to be uniform. The lines of response are rays of a finite width which depends on the detectors' resolution. The "coincidence counts" for each line of response are proportional to the line integral of the tracer concentration. The Radon transform operator, R[f(x,y)] defines the integral of f(x,y) along a line, and analytical methods approximate the inverse Radon operator to determine the function f.
Filtered Back Projection (FBP) is an analytical PET image reconstruction method in which the emission data is first frequency filtered and is then back-projected into the image space. Back-projection of unfiltered emission data, p, is known as simple back-projection. Back-projection of p assigns a value which is proportional to the number of coincidence counts for a line of projection to each pixel through which that line passes. The resulting image is a star like pattern, with pixel values peaking at the position of the source. This means that nearby pixels will have non-zero values although no tracer is present at that location. The star-like image may be significantly reduced by initial frequency filtering of the emission data. The peak is principally composed of high-frequency components, while areas outside the peak are composed of low-frequency components. High-pass frequency filtering the emission data amplifies the high-frequency components relative to the low-frequency components, and the back-projection of the filtered projection data is of higher visual quality.
Iterative methods of reconstruction require that the emission data is discretized before solution of the reconstruction problem. A finite-dimensional algebraic system of linear equations are solved by some iterative method. Iterative methods of reconstruction are more flexible than analytical methods because weights or penalties which reflect the nature of the problem may be introduced into the set of linear equations. For example, additional constraints and penalty functions can be included which ensure that the solution has certain desired properties, such as requiring that values of neighboring pixels should be close to one another.
Iterative PET image reconstruction methods may be divided into three classes: Norm Minimizations, Maximum Likelihood Approaches, and Maximum A Posteriori Approaches. The prior art includes many Norm Minimization methods, although those which involve factorization of the matrix are usually unsuitable because of the size of the computational power required for solution. One set of norm minimization method, Algebraic Reconstruction Techniques, or Row Action Methods, are relaxation methods which require access to only one row of matrix per iteration. Maximum Likelihood Approaches (MLAs) maximize the conditional probability of observing the coincidence counts over all possible image vectors. MLA models which assumes the emission process has a Poisson distribution are considered to be the most appropriate to PET reconstruction, and are usually solved by the expectation maximization (EM) iterative method. Maximum A Posteriori Approach (MAPA) models maximize the conditional probability of the image vector given the measurement vector or observed coincidences. MAPA models require that the a priori probability distribution of the images is known, which has been considered a drawback. A recent comparison of the various iterative methods of PET image reconstruction concluded that a Maximum Likelihood Approach using EM methods best characterized the emission process and were the most promising direction for further PET image reconstruction research.